We consider the problem of minimizing a function f of n real variables $x_1 , \cdots ,x_n $, subject to the provision that derivative information is not to be used in seeking the minimum. A number of methods exist for solving this problem and of these, several are based on the construction of conjugate or orthogonal search directions. The aim of this paper is to demonstrate how these methods may be adapted to work when linear constraints on the variables are present. We describe how orthogonal transformations may be applied to the search directions so that conjugacy or orthogonality relations are preserved whenever the search directions are modified in order to ensure that no constraints are violated.